On the Cauchy problem for a linear harmonic oscillator with pure delay

Khusainov, Denys Ya; Pokojovy, Michael; Azizbayov, Elvin I.

On the Cauchy problem for a linear harmonic oscillator with pure delay

Files

Khusainov_0-296855.pdf(1.6 MB)

Date

2015

Authors

Khusainov, Denys Ya

Pokojovy, Michael

Azizbayov, Elvin I.

Publication type

Journal article

Published in

Advances in Difference Equations ; 2015 (2015). - 197. - ISSN 1687-1839. - eISSN 1687-1847

Abstract

In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace so that the solution of the homogeneous problem can be written as a linear combination of these two eigenfunctions. As opposed to that, in the presence even of a small delay, the spectrum is infinite and a finite sum representation is not possible. Using a special function referred to as the delay exponential function, we give an explicit solution representation for the Cauchy problem associated with the linear oscillator with pure delay. Finally, the solution asymptotics as the delay parameter goes to zero is studied. In contrast to earlier works, no positivity conditions are imposed.

Subject (DDC)

510 Mathematics

Keywords

functional-differential equations, harmonic oscillator, pure delay, well-posedness, solution representation

Cite This

ISO 690

KHUSAINOV, Denys Ya, Michael POKOJOVY, Elvin I. AZIZBAYOV, 2015. On the Cauchy problem for a linear harmonic oscillator with pure delay. In: Advances in Difference Equations. 2015, 197. ISSN 1687-1839. eISSN 1687-1847. Available under: doi: 10.1186/s13662-015-0538-z

BibTex

@article{Khusainov2015Cauch-32063,
  year={2015},
  doi={10.1186/s13662-015-0538-z},
  title={On the Cauchy problem for a linear harmonic oscillator with pure delay},
  volume={2015},
  issn={1687-1839},
  journal={Advances in Difference Equations},
  author={Khusainov, Denys Ya and Pokojovy, Michael and Azizbayov, Elvin I.},
  note={Article Number: 197}
}

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